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Minimax Theory and its Applications 01 (2016), No. 1, 051--063
Copyright Heldermann Verlag 2016

Representation of Viscosity Solutions of Hamilton-Jacobi Equations

Emmanuel N. Barron
Department of Mathematics and Statistics, Loyola University, Chicago, IL 60660, U.S.A.


Hamilton Jacobi equations of the form $H(x,u,Du)=0$ are considered with $H(x,r,p)$ nondecreasing in $r$ and quasiconvex in $p$. A viscosity solution may be represented as the value function of a calculus of variations or control problem in $L^\infty$, i.e., as a minimax problem. For time dependent problems of the form $u_t+H(t,x,u,Du)=0$ we require that $H(t,x,r,p)$ is convex in $p$ and nondecreasing in $r$. The viscosity solution is then given as the value of an $L^\infty$ problem.

Keywords: Quasiconvex, Hamilton-Jacobi, representation.

MSC: 35F21, 49L25

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